Chebyshev collocation spectral method to solve radiative transfer equation in one-dimensional cylindrical medium

Abstract

A Chebyshev collocation spectral method (CCSM) is developed to solve the radiative transfer equation (RTE) in an infinitely long, cylindrically symmetric, homogeneous medium. Both the spatial and angular computational domains of the RTE are discretized by the Chebyshev collocation points. For the CCSM, taking the conservation form of RTE, which is commonly used in the discrete ordinates method (DOM), will produce poor accuracy, whereas taking the non-conservation form can yield good prediction. To test the applicability of different collocation point schemes in the radial direction, three solvers are developed. SOLVER1 and SOLVER2 use the Chebyshev-Gauss-Radau (CGR) points and the Chebyshev-Gauss-Lobatto (CGL) points on the radius, respectively. SOLVER 3 uses the CGL points on the diameter, and an even number of nodes is taken to exclude the origin. The results show that SOLVER1 and SOLVER2 suffer from the “singularity effect”. This effect can be reduced by increasing the grid number in radial direction. Whereas SOLVER3 can avoid the “singularity effect”. We also show that the mapping (the Kosloff–Tal-Ezer transformation) actually cannot improve the accuracy. Besides, the numerical accuracy is declined by using cosine of the polar angle instead of the polar angle itself as the independent variable. Our results demonstrate that the CCSM performs much better than the DOM and produces the exponential convergence in both the spatial and angular domains. The CCSM is a superior method to achieve high accuracy for thermal radiation problem of homogeneous medium with cylindrical symmetry.